Note that there exist a couple of different stationarity concepts like
W-, C-, M-, and S-stationarity for a number of related problem classes,
including mathematical programs with complementarity constraints [1]},
cardinality constraints [2]}, vanishing constraints [3]},
and switching constraints [4]}.
Similarly, it would be possible to state some of these other stationarity concepts
for problem REF  as well. However, on the one hand, it turns out
that suitable methods for the solution of sparse optimization problems can
be shown to converge to S-stationary points, see [5]} for some preliminary
results in this direction, which is in contrast to the other classes of
problems mentioned before and which indicates that there is no need to introduce
these weaker stationarity concepts for sparse optimization problems, and, on the other
hand, for the purpose of the approach presented here, we only require the
S-stationarity from Definition REF .
